\frame{ \frametitle{Method: Principal Component Analysis}
  \begin{itemize}
   \item First, a
    \hyperlink{pca}{\beamergotobutton{principal component analysis}}
    is done on $S_1$ and $S_2$ to yeild a dimensionality of $d < N$.
  \vspace{8pt}
  \begin{center}
  \includegraphics[width=.4\textwidth]{pca}
  \end{center}
  \end{itemize}
}

\frame{ \frametitle{Method: Mapping to Grassmann Manifold}
  \begin{itemize}
   \item Then, the points are mapped to the
    \hyperlink{grassmann-manifold}{\beamergotobutton{Grassmann manifold}}
    $\mathbb{G}_{N,d}$.
  \vspace{8pt}
  \begin{center}
  \includegraphics[width=.7\textwidth]{grassmann-manifold}
  \end{center}
  \end{itemize}
}

\frame{ \frametitle{}
  \begin{itemize}
   \item The objective is to find a
    \hyperlink{geodesics}{\beamergotobutton{geodesic}}
    between $S_1$ and $S_2$ from which the intermediate subspaces may be
    derived.
  \vspace{8pt}
  \begin{center}
  \includegraphics[width=.7\textwidth]{geodesic}
  \end{center}
  \end{itemize}
}

\frame{ \frametitle{}
  \includegraphics[width=\textwidth]{domain-adaptation}
}

\frame{ \frametitle{The Geodesic Path}
  \begin{itemize}
   \item Then the geodesic path is computed as:
     \begin{equation}
      \Psi(t') = Q exp(t'B) J,
     \end{equation}
     where exp() is the
    \hyperlink{matrix-exponential}{\beamergotobutton{matrix exponential}},
    $t'$ is the time parameter, and
     \begin{equation}
      B = \begin{bmatrix} 0 & -A \\ -A & 0 \end{bmatrix},
     \end{equation}
     \begin{equation}
      Q^T S_1 = J = \begin{bmatrix} I_d \\ 0_N-d,d \end{bmatrix}.
     \end{equation}
  \end{itemize}
}

\frame{ \frametitle{}
  \includegraphics[width=\textwidth]{geodesic-path}
}

\frame{ \frametitle{The Inverse Exponential Map}
  \begin{itemize}
   \item The algorithm for computing geodesic flow, given $S_1$ and
   $S_2$, is:
   \begin{enumerate}
    \item Compute the $N x N$
    \hyperlink{orthogonal-completion}{\beamergotobutton{orthogonal completion}}
    $Q$ of $S_1$.
    \item Compute the
    \hyperlink{cs-decomposition}{\beamergotobutton{CS decomposition}}
    of $Q^TS_2$, given by:
    \begin{equation}
     \begin{pmatrix} V_1 & 0 \\ 0 & V_2    \end{pmatrix}
     \begin{pmatrix} \Gamma(1) \\ -\Sigma(1) \end{pmatrix}
     V^T
    \end{equation}
    \item Compute $\{\theta_i\}$ by taking the arccos and arcsin of
          $\Gamma$ and $\Sigma$, respectively, and form the diagonal
          matrix $\Theta$.
    \item Finally, $A = \bar{V}_2 \Theta V_1^T$.
   \end{enumerate}
  \end{itemize}
}

\frame{ \frametitle{The Exponential Map}
  \begin{itemize}
   \item The algorithm for computing geodesic flow, given
   a tangent vector
   $B = \begin{bmatrix} 0 & A^T \\ A & 0 \end{bmatrix}:$
   \begin{enumerate}
    \item Compute the $N x N$
    \hyperlink{orthogonal-completion}{\beamergotobutton{orthogonal completion}}
    $Q$ of $S_1$.
    \item Compute the compact
    \hyperlink{svd}{\beamergotobutton{SVD}}
    of $A$.
    \item Compute the diagonal matrices $\Gamma(t')$ and $\Sigma(t')$
          such that
          $\gamma_i(t') = cos(t'\theta_i)$ and
          $\sigma_i(t') = sin(t'\theta_i)$, where $\theta_i$ is the $i^{th}$
          diagonal element of $\Theta$.
    \item Finally, compute $\Psi(t') = Q
                 \begin{pmatrix}
                    V_1 \Gamma(t') \\
                    V_2 \Sigma(t')
                 \end{pmatrix}$,
          for various values of $t' \in [0,1]$.
   \end{enumerate}
  \end{itemize}
}

\frame {\frametitle {Performing Recognition Under Domain Shift}
    Train a classifier D(Xl¡¯, Yl¡¯) using the data from the subspaces.
  \begin{figure}
    \includegraphics[width = 0.95 \textwidth] {3_4_image1}
  \end{figure}
}

\frame { \frametitle {}
  \begin{figure}
    \includegraphics[width = 0.95 \textwidth] {3_4_image2}
  \end{figure}
}
